Some parametric congruences involving generalized central trinomial coefficients

Abstract

For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central trinomial coefficient. In this paper, we mainly establish some parametric congruences involving generalized central trinomial coefficients. As consequences, we prove that for any prime $p>3$ $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}}{12^k}T_k\equiv\left(\frac{p}{3}\right)\frac{3^{p-1}+3}{4}\pmod{p^2} $$ and $$ \sum_{k=0}^{p-1}\frac{T_kH_k}{3^k}\equiv\frac{3+\left(\frac{p}{3}\right)}{2}-p\left(1+\left(\frac{p}{3}\right)\right)\pmod{p^2}, $$ where $(-)$ denotes the Legendre symbol and $H_k:=\sum_{j=1}^k1/j$ denotes the $k$th harmonic number. These confirm two conjectural congruences of the second author.